# Two step (spherical) square well potential

## Homework Statement

Just need some rough guidance on this one, nothing specific is really needed. The problem:

Given a spherically symmetric potential (V(r))
$$V\left( r \right) = \left\{\begin{gathered} V_0 \hfill \hspace{2}r<a \\ 0 \qquad a<r<b \\ \infty \hfill r>b \\ \end{gathered} \right$$

Find the energies for the ground state and the first excited state. Also find an (approximate) expression for the energy splitting of the levels if $V_0$ is very large compared to these energy levels.

## Homework Equations

Must use spherical Bessel and Neumann functions.

## The Attempt at a Solution

The wavefunctions:

$$\mathcal{U}_I(r)=A_rJ_{\ell}(k_1r)$$
$$\mathcal{U}_{II}(r)=C_rJ_{\ell}(k_2r)+D_r n_{\ell}(k_2r)$$
$$\mathcal{U}_{III}(r)=0$$

$$k_I=\frac{\sqrt{2m(V_0-E)}}{\hbar}$$

$$k_{II}=\frac{\sqrt{2mE}}{\hbar}$$

where the spherical Neumann function ($$n_{\ell}$$) goes away in the first function since it is singular at the origin.

At this point work with the following boundary conditions:

$$\mathcal{U}_I(a)=\mathcal{U}_{II}(a)$$
$$\mathcal{U}_I^{'}(a)=\mathcal{U}_{II}^{'}(a)$$
$$\mathcal{U}_II(b)=0$$

From here you get a three equations, where the derivatives are taken with respect to K1 and K2 (respectfully). First question: How do I take a derivative of the spherical Bessel and Neumann functions? Should I use the recursion formula and solve for the derivative?

Actually I am just not quite sure where to go from here in terms of finding the ground state energy. If this were a regular potential (two step) barrier, all I would do is eliminate the three constants using the three equations, arriving at a transcendental equation which I would then graph graph each side of the equation separately and find the intersections which represent the energy eigenvalues. Same thing here? Any thoughts???

Thanks yall

IHateMayonnaise